As far as I know there is no designated burial ground within Hawkhead (Leverndale) Hospital grounds. The grounds were extensive and included workshops, farms & Orchards, a laundry and a golf course. (My mother was a nurse there- and sometimes cajoled me into working at the Xmas parties). - it is possible that there had been a practice of burials in house so to speak but certainly by 1920 I think this practice would have ceased. Hawkhead Cemetry in Paisley opened in 1891 and could be reached quite easily from the hospital grounds - although the boundary problem rears its head there.
The Merryflats connection does explain why he ended up in a Glasgow hospital but as you have hinted at the stigma of Hawkhead may have had quite a detrimental effect on the family.
Given that he was a military transfer it is possible that his records were swept up when the MOD gathered all records of personnel treated in local and temporary war hospitals sometime in the 1920's. In a previous thread it was discovered that the NHS archivist had no records of Merryflats ever having been used as a Military Hospital.
There will be a record somewhere of the burial - even if it is just the Parish/Council keeping track of their expenditure - the problems lie in finding it
